#134444

order	512
nilpotency class	4
isoclinism	isoclinic class of #134444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7,8,5)(4,6)(10,16)(11,17,20,24)(12,14)(13,18)(19,22,21,23);(9,10,12,16,15,18,14,13)(11,17,19,23,20,24,21,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,19)(13,22)(14,21)(15,20)(16,23)(18,24)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41640);